As many others have already noted, the Tricki project is now on the verge of going live, thanks to the efforts of Tim Gowers, Alex Frolkin, and Olof Sisask. This seems to be a sign among many that the Internet can and will have a profound impact on how mathematical research is done, and it is intriguing to speculate about how communication technology will change the way we do mathematical research in the coming decades (and centuries).
The Internet has of course already changed a lot of things, many of which we already take for granted. We enjoy the advantages of preprint servers such as the arXiv. Lots of basic mathematical knowledge is instantly available at Wikipedia, and I was recently very happy to discover the nLab. Some people are writing a book about stacks in an online collaborative project. Others create resource pages on specific subjects, like motivic homotopy theory. Needless to say, there are lots of math blogs, and lots of online books and lecture notes. Tim Gowers once proposed a site with alternative maths reviews. There are various useful databases, like the Sloane’s Online Encyclopedia of Integer Sequences and John Cremona’s tables of elliptic curves, both available through SAGE. Although the choice of subject matter prevented me from taking part, I very much liked the idea of the polymath experiment.
So what will the future bring? All mathematical definitions implanted in a chip in your brain? Quantum computers emulating the brain of Grothendieck? Computers actually inventing new mathematics? One can only guess about these and other developments, but in the shorter run it seems to me that a key question will be to find a way to make all of the mathematics literature available online, for free. Having been blogging for a few months now, it is very impractical to not being able to link to articles, just because the author has given away the copyright to a huge profit-hungry company, or because the article only exists in paper form. The same problem must face anyone trying to implement any form of open online collaborating. Some very clever people developed Spotify for music lovers, making almost all music available for free while keeping the music industry happy. Who will do the same for maths lovers, and for other scientists?
Motivic stuff 